Permutation

A **permutation** is a collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter. In other words, a **permutation is an arrangement of objects in a definite order**. So before deep dive into permutation let’s have a brief discussion on factorial first.

# Factorial

- Factorial of a natural number n is denoted by the notation n!
- n! is the product of all natural numbers starting from 1 till n, including 1 and n.

i.e. n × (n-1) × (n-2) × (n-3) . . . × 1

# Permutation

- If n objects are available and we arrange all, then every arrangement possible is called a permutation.
- If out of n available objects, we choose r and arrange them. Then every possible arrangement is called an r — permutation.
- In permutation the order of objects matters.

In permutation, we primarily deal with four kinds of problems

**Permutation with repetition****Permutation without repetition****r — permutation without repetition****r- permutation with repetition**

# Permutation Formula ( nPr )

From the previous example, we understood thatr

nPr = n × (n — 1) . . . (n — r + 1)Multiplying and Dividing by (n — r)!

nPr = n × (n — 1) × (n — 2) × . . . × (n — r + 1) × (n — r)! / (n — r)!nPr = n × (n — 1) × (n — 2) × . . . × (n — r + 1) × (n — r) × (n — r — 1) × . . . × 1 / (n — r)!

nPr = n! / (n — r)! where 0 ≤ r ≤ n